As of this writing (and much is surely still up in the air), Napster has lost a major legal battle and is seeking to settle the suit by offering to pay $1 billion out of their future profits. In order to pay those fees, Napster is planning to institute a monthly fee for users.
Napster's technology requires the use of their central servers, which undoubtably made them a convenient legal target. It is unclear whether fully peer to peer (no central server) systems such as Gnutella will be on the availability of free music sharing on the Internet. Surely if Gnutella develops into a viable (and free alternative), but with no central server to take the legal heat, Napster's proposed business model (paid access) will fail.
Another potential competitor or successor to Napster is OpenNAP. Since most existing Napster clients can select what server they connect to, and since there is a free software clone of Napster server, anyone on the Internet can theoretically open up a small competitor or successor to Napster. If this is true, then the death of Napster might only mean a temporary slowdown in the growth of filesharing.